Intra-individual versus inter-individual correlation
This article deals with the difference between the intra-individual and inter-individual correlation. These correlations are associated with different types of variation that stem from different types of data. The difference between these correlations often forms the basis for the ecological fallacy, and it forms an important reason for doing research with intensive longitudinal data (ILD), rather than cross-sectional of panel data.
Specifically, the intra-individual correlation is based on intra-individual or within-person variation that is characteristic of N=1 time series data. It can also be considered within the context of N>1 ILD, as such data can be thought of as a collection of N=1 time series. In contrast, the inter-individual correlation is based on inter-individual variation, which is typically associated with cross-sectional data. However, inter-individual variation and the inter-individual correlation can also be considered in the context of panel data or ILD, as will be explained in this article. For this reason, the current article refers to the intra-individual versus inter-individual correlation, rather than the within-person versus cross-sectional correlation.
In this article, you will find: 1) an explanation of how the intra-individual and inter-individual correlations are based on different deviations; and 2) a discussion of how the auto-correlation, that is, the correlation between a variable and itself measured at an earlier occasion, can also be based on either intra- or inter-individual variation, and how these correlations can be interpreted. For the exact mathematical relation between these correlations, you can see Hamaker (2024); here the focus is on obtaining an intuitive understanding of the origin of the difference between these two correlations.
1 Different correlations based on different deviations
Correlations are often used as descriptive measures to quantify the strength of a relation between two variables: They are easy to interpret, because they always lie between -1 and 1, and a correlation of 0 implies there is no (linear) relation. Moreover, many of the analysis techniques that are used in psychology and related fields are based on analyzing the covariance structure of the data, and since correlations are a standardized form of this, they are central to many analysis techniques as well.
You can describe a correlation as a way to assess how deviations from the mean in one variable are related to deviations from the mean in another variable. It is thus critical which mean is being considered: Depending on whether the intra-individual (i.e., within-person) mean or the inter-individual mean is being used, the correlation will be based on intra-individual or inter-individual variation.
Below you can read more on the way an intra-individual correlation is obtained, and how the inter-individual correlation is obtained.
1.1 Intra-individual correlation
When you have N=1 time series data, which consist of observations at many occasions for a single person (or other case, such as a dyad), you can consider the temporal deviations of this person from their own long-run mean over time. By relating these intra-individual or within-person deviations on two variables to each other, you obtain the intra-individual or person-specific correlation.
Agda wants to see whether Kalle is more nervous on days when he drinks more coffee compared to days when he drinks less coffee. To this end, she uses a daily diary study to obtain measures of daily coffee consumption and nervousness from Kalle on a large number of days.
With these data Agda computes Kalle’s correlation between these variables. She expects this intra-individual correlation to be positive, indicating that Kalle tends to be more nervousness on days when his coffee consumption was higher.
The intra-individual correlation is thus concerned with intra-individual or within-person variation. It may be different for different people: One person may be characterized by a strong intra-individual correlation, while for another person the intra-individual correlation is only weak. It is also possible that the sign of the intra-individual correlation differs across individuals. You may actually consider individual differences in the intra-individual correlation as your research question.
1.2 Inter-individual correlation
A much more common correlation to encounter in the psychological literature is the inter-individual correlation. It is typically based on cross-sectional data that consist of observations of many individuals at a single occasion. To obtain the inter-individual correlation, you have to consider the deviation of each person’s score from the grand mean of the group; it thus captures how such inter-individual deviations on two variables are related to each other.
Chen wants to know whether people who drink more coffee feel more nervous than people who drink less coffee. He therefore wants to obtain data from a large sample of individuals, measuring their daily coffee consumption and their nervousness.
With these data, Chen can compute the inter-individual correlation. A positive correlation would imply that individuals who drank more coffee, reported more nervousness.
The inter-individual correlation is thus concerned with inter-individual variation, which are often referred to as individual differences. This correlation may also vary over time, which is something you can further investigate when you obtain panel data.
1.3 Visualizing the various deviations
To better appreciate the difference between the intra-individual and the inter-individual correlations, it can be helpful to use visualizations of the deviations that they are based on. Consider the five observations of a single person represented by the grey dots in Figure 1. The mean of this person is represented by the dashed blue line; it is the intra-individual, also referred to as the within-person or person-specific mean.
Let’s focus on the second observation: It lies below the person’s mean (represented by the dashed blue line), which implies that the deviation between this observation and the person’s mean is negative. This deviation is represented in red; it is deviations like these that are used when computing the intra-individual correlation.
Now consider this same observation, but instead of it being part of an N=1 dataset, let’s assume it is obtained as part of a cross-sectional data set. In this case, the same observation is not compared to the intra-individual mean, but rather to the inter-individual or cross-sectional mean, as shown in Figure 2 (for simplicity, it is assumed here that the cross-sectional mean does not change over time, so it can also be referred to as the grand mean).
Hence, although the observation itself is not different then before, the mean to which it is compared is now different. As a result, the deviation is also different: Where it was negative before (when compared to the intra-individual mean), it is now positive, represented in purple.
In Figure 3, the two means and the deviations they result in are included in a single plot. It also shows the deviation of the intra-individual mean from the grand mean (in blue); this deviation is referred to as the between-person deviation, as it captures only between-person variation.
What you can see from Figure 3 is that the purple inter-individual deviation (which is the basis for the inter-individual correlation), is actually the sum of the relatively large between-person deviation in blue and the negative and smaller intra-individual deviation in red. For other people in the same cross-section other combinations are possible.
For instance, the intra-individual deviation can be positive and large, whereas the between-person deviation is also positive, but less big; then, the inter-individual deviation will be positive and very large. Alternatively, the intra-individual deviation can be large and positive, while the between-person deviation is large and negative; as a result the two may almost cancel each other out, so that the inter-individual deviation ends up to be close to zero.
1.4 Conclusion
The difference between the intra-individual and inter-individual correlations comes from the fact that the two are based on different types of data, and that these data are characterized by different means. As a result, the same observation will typically have a different deviation from the intra-individual mean than from the grand mean, thereby resulting in different contributions to these correlations.
This implies that the intra-individual correlation and the inter-individual correlation can be quite different from each other. It is for instance possible that the intra-individual correlation of all people in a population is negative, but that the cross-sectional correlation is positive.
Chen has found that cross-sectionally, there is no relation between daily coffee consumption and nervousness. When discussing this result with Agda, she points out that in her daily diary study, she has now studied the relation between daily coffee consumption and nervousness in multiple individuals separately, and she has found that all of the participants are characterized by a positive intra-individual correlation.
Over coffee Chen and Agda discuss where the seemingly contradictory results may stem from, and realize that there is a third kind of correlation that neither of them has studied: The between-person correlation based on inter-individual differences in the intra-individual means on coffee consumption and nervousness. If they obtain enough occasions from enough people, they can actually use an analysis technique that allows them to look into this correlation as well.
Differences in the sign of intra-individual and inter-individual correlations can occur under certain circumstances; this is further explained in the article about the [within/between problem]. The exact mathematical relation between the intra-individual correlation and inter-individual correlation is discussed in Hamaker (2024). Furthermore, you can consider this Shiny app, which forms an interactive tool that allows you to get more insight in the cross-sectional correlation.
2 Intra- and inter-individual autocorrelations
Instead of considering the correlation between two different variables measured at the same time, which gives you a concurrent or contemporaneous correlation as described above, you can also consider the correlation between variables measured at different occasions. If these are different variables measured at different occasions, the resulting correlation can be referred to as a [cross-lagged correlation]. An example of this could be the correlation between yesterday’s physical activity and today’s mood. Just like the concurrent correlations, lagged cross-correlations can also be obtained based on intra-individual or inter-individual variation.
A particularly interesting lagged correlation is that between a variable and itself. This is referred to as the [autocorrelation]; it can also be based on both intra-individual and inter-individual variation, as explained below.
2.1 Intra-individual autocorrelation: Carry-over or inertia
When you obtain N=1 time series data, you can compute the correlation between a variable and itself at a particular time interval. For instance, when you have daily diary data, you can compute the correlation between a variable and itself a day earlier, or two days earlier. Similarly, when you have physiological measurements such as heart rate, you can compute the correlation at a 10 second interval, or a one minute interval.
The distance between these occasions is referred to as the lag, and is typically expressed in terms of the number of time steps on the timescale at which the measures are obtained. For instance, when we have daily diary data, a lag of 1 is referring to a distance in time of 1 day.
Agda wants to know whether the daily levels of anxiety experienced by Kalle fluctuate randomly over time, or that there is some pattern in these fluctuations. She therefore decides to obtain daily diary data, and to correlate every day’s anxiety score reported by Kalle with the anxiety he reported on the preceding day. This way, she can compute the lag 1 autocorrelation: It can be described as the correlation between today and yesterday’s anxiety.
She finds an autocorrelation of 0.36, meaning that if Kalle experiences more anxiety than is usual for him, this tends to also be true to some extent the next day; similarly, if he experiences less anxiety than usual, this also tends to be true the following day.
Agda also obtained daily diary measures of her own anxiety levels. In her own data, she finds an autocorrelation close to zero. This implies that for Agda, there is no carry-over of her anxiety level from one day to the next. Her dynamic pattern can thus be described as “every day is a new day”.
The intra-individual autocorrelation at lag 1 reflects the degree to which the measured process tends to remain above or below its own mean from one occasion to the next. It is therefore also referred to in some parts of the psychological literature as inertia (Gottman et al., 1999; Suls et al., 1998). Other terms that have been used to describe this dynamic feature are carry-over, as it represent the degree to which a score carries over to the next occasion, and regulatory weakness, as it forms a measure of how much time it takes to return to one’s mean after having moved away from this.
It is also possible for an autocorrelation to be negative; in that case, we would not refer to it as inertia, carry-over, or regulatory weakness, but rather as a dynamic feature that represents some kind of feedback process or compensating behavior, as high scores tend to be followed by low scores, and vice versa.
2.2 Inter-individual autocorrelation: Test-retest correlation or stability
In the context of inter-individual research, we can also consider autocorrelations; these are more commonly referred to as test-retest correlations. Such correlations of individual differences on the same variable at different measurement occasions are often interpreted as a way to quantify the stability of the construct that is being measured, or to investigate the reliability of the construct.
Chen wants to know whether individual difference in drinking tend to remain stable over time. He therefore obtains measurements of daily alcohol consumption from a large sample of people at two occasions, and computes the correlation between these measures. The test-retest correlation he obtains is 0.47, indicating that there is some stability, but that there is also considerable instability over time.
To determine whether this instability reflects genuine fluctuations in the construct itself, or that it results from measurement error, Chen considers looking into approaches that can be help to determine the reliability of single item measures.
A test-retest correlation close to 1 implies that the rank-order of individuals remains largely the same from one measurement occasion to the next; in contrast, a test-retest correlation close to zero implies that the ordering of individuals has changed substantially, and knowing the prior score tells us little to nothing about what to expect at the follow-up measurement occasion.
3 Think more about
The degree to which an observation reflects a momentary state or a more enduring feature of an individual, will depend in part on the time frame of your measurement instrument, that is, the period to which an observation pertains. For instance, when being asked how tired you are right now versus how tired you have been over the past week, you are asked to make inherently different evaluations that are likely to result in different answers: You may feel quite tired now because you did not sleep well, but when considering the entire past week you may conclude that on average you were not particularly tired.
This implies that the correlations you compute, whether intra-individual or inter-individual, and whether concurrent or lagged, depend on the time frame that you use. Moreover, when considering lagged correlations, whether between different variables or the same variable, the time interval between your measurements will also play a critical role in the result you obtain.
Hence, when interpreting correlations, it is important to realize whether they pertain to variation within a person or variation across individuals. Interpreting one as representing the other is known as an instance of the ecological fallacy. Only when the conditions of ergodicity are met, can you make such generalizations from the individual level to the population and back without any danger of bias.
4 Takeaway
When computing a correlation, you first have to determine for each observation how it deviates from the mean. However, you can consider different means for this, most notably, the person’s specific intra-individual mean, or the grand mean. When these means are different, the deviations of a score to each of these means will be different as well. As a result, the correlations that are computed based on these deviations can differ substantially.
This is true for the concurrent correlation (i.e., between variables measured at the same occasion), but also for lagged correlations (i.e., between variables measured at different points in time). A specific form of the latter are autocorrelations, which are based on correlating a variable with itself at another occasion.
Hence, it is important to realize that there is not such a thing as “the” correlation between two variables, or “the” auto-correlation when considering a single variable; it always depends critically on how the measures were obtained and how the correlation was computed, and therefore you should include this in your interpretation and description of the results.
5 Further reading
We have collected various topics for you to read more about below.
- [Intra- versus inter-individual variation]
- Ergodicity in psychology
- [The within/between problem]
- Ecological fallacy
Acknowledgments
This work was supported by the European Research Council (ERC) Consolidator Grant awarded to E. L. Hamaker (ERC-2019-COG-865468).
References
Citation
@article{hamaker2025,
author = {Hamaker, Ellen L. and Hoekstra, Ria H. A.},
title = {Intra-Individual Versus Inter-Individual Correlation},
journal = {MATILDA},
number = {2025-05-23},
date = {2025-04-23},
url = {https://matilda.fss.uu.nl/articles/intra-vs-inter-individual-correlation.html},
langid = {en}
}